Robust Statistics: Theory and MethodsISBN: 9780470010921
436 pages
May 2006

Robust Statistics sets out to explain the use of robust methods and their theoretical justification. It provides an uptodate overview of the theory and practical application of the robust statistical methods in regression, multivariate analysis, generalized linear models and time series. This unique book:
 Enables the reader to select and use the most appropriate robust method for their particular statistical model.
 Features computational algorithms for the core methods.
 Covers regression methods for data mining applications.
 Includes examples with real data and applications using the SPlus robust statistics library.
 Describes the theoretical and operational aspects of robust methods separately, so the reader can choose to focus on one or the other.
 Supported by a supplementary website featuring timelimited SPlus download, along with datasets and SPlus code to allow the reader to reproduce the examples given in the book.
Robust Statistics aims to stimulate the use of robust methods as a powerful tool to increase the reliability and accuracy of statistical modelling and data analysis. It is ideal for researchers, practitioners and graduate students of statistics, electrical, chemical and biochemical engineering, and computer vision. There is also much to benefit researchers from other sciences, such as biotechnology, who need to use robust statistical methods in their work.
1. Introduction.
1.1 Classical and robust approaches to statistics.
1.2 Mean and standard deviation.
1.3 The “threesigma edit” rule.
1.4 Linear regression.
1.5 Correlation coefficients.
1.6 Other parametric models.
1.7 Problems.
2. Location and Scale.
2.1 The location model.
2.2 Mestimates of location.
2.3 Trimmed means.
2.4 Dispersion estimates.
2.5 Mestimates of scale.
2.6 Mestimates of location with unknown dispersion.
2.7 Numerical computation of Mestimates.
2.8 Robust confidence intervals and tests.
2.9 Appendix: proofs and complements.
2.10 Problems.
3. Measuring Robustness.
3.1 The influence function.
3.2 The breakdown point.
3.3 Maximum asymptotic bias.
3.4 Balancing robustness and efficiency.
3.5 *“Optimal” robustness.
3.6 Multidimensional parameters.
3.7 *Estimates as functionals.
3.8 Appendix: proofs of results.
3.9 Problems.
4 Linear Regression 1.
4.1 Introduction.
4.2 Review of the LS method.
4.3 Classical methods for outlier detection.
4.4 Regression Mestimates.
4.5 Numerical computation of monotone Mestimates.
4.6 Breakdown point of monotone regression estimates.
4.7 Robust tests for linear hypothesis.
4.8 *Regression quantiles.
4.9 Appendix: proofs and complements.
4.10 Problems.
5 Linear Regression 2.
5.1 Introduction.
5.2 The linear model with random predictors 118
5.3 Mestimates with a bounded ρfunction.
5.4 Properties of Mestimates with a bounded ρfunction.
5.5 MMestimates.
5.6 Estimates based on a robust residual scale.
5.7 Numerical computation of estimates based on robust scales.
5.8 Robust confidence intervals and tests for Mestimates.
5.9 Balancing robustness and efficiency.
5.10 The exact fit property.
5.11 Generalized Mestimates.
5.12 Selection of variables.
5.13 Heteroskedastic errors.
5.14 *Other estimates.
5.15 Models with numeric and categorical predictors.
5.16 *Appendix: proofs and complements.
5.17 Problems.
6. Multivariate Analysis.
6.1 Introduction.
6.2 Breakdown and efficiency of multivariate estimates.
6.3 Mestimates.
6.4 Estimates based on a robust scale.
6.5 The Stahel–Donoho estimate.
6.6 Asymptotic bias.
6.7 Numerical computation of multivariate estimates.
6.8 Comparing estimates.
6.9 Faster robust dispersion matrix estimates.
6.10 Robust principal components.
6.11 *Other estimates of location and dispersion.
6.12 Appendix: proofs and complements.
6.13 Problems.
7. Generalized Linear Models.
7.1 Logistic regression.
7.2 Robust estimates for the logistic model.
7.3 Generalized linear models.
7.4 Problems.
8. Time Series.
8.1 Time series outliers and their impact.
8.2 Classical estimates for AR models.
8.3 Classical estimates for ARMA models.
8.4 Mestimates of ARMA models.
8.5 Generalized Mestimates.
8.6 Robust AR estimation using robust filters.
8.7 Robust model identification.
8.8 Robust ARMA model estimation using robust filters.
8.9 ARIMA and SARIMA models.
8.10 Detecting time series outliers and level shifts.
8.11 Robustness measures for time series.
8.12 Other approaches for ARMA models.
8.13 Highefficiency robust location estimates.
8.14 Robust spectral density estimation.
8.15 Appendix A: heuristic derivation of the asymptotic distribution of Mestimates for ARMA models.
8.16 Appendix B: robust filter covariance recursions.
8.17 Appendix C: ARMA model statespace representation.
8.18 Problems.
9. Numerical Algorithms.
9.1 Regression Mestimates.
9.2 Regression Sestimates.
9.3 The LTSestimate.
9.4 Scale Mestimates.
9.5 Multivariate Mestimates.
9.6 Multivariate Sestimates.
10. Asymptotic Theory of Mestimates.
10.1 Existence and uniqueness of solutions.
10.2 Consistency.
10.3 Asymptotic normality.
10.4 Convergence of the SC to the IF.
10.5 Mestimates of several parameters.
10.6 Location Mestimates with preliminary scale.
10.7 Trimmed means.
10.8 Optimality of the MLE.
10.9 Regression Mestimates.
10.10 Nonexistence of moments of the sample median.
10.11 Problems.
11. Robust Methods in SPlus.
11.1 Location Mestimates: function Mestimate.
11.2 Robust regression.
11.3 Robust dispersion matrices.
11.4 Principal components.
11.5 Generalized linear models.
11.6 Time series.
11.7 Publicdomain software for robust methods.
12. Description of Data Sets.
Bibliography.
Index.
Doug Martin is a Professor in the Department of Statistics, and Director of the Computational Finance Program at the University of Washington in Seattle, Washington. He was a consultant at Bell Laboratories for many years, and author of numerous research articles on robust methods for time series. Martin founded the original SPLUS company Statistical Sciences, Inc., and led the development of the SPLUS Robust Statistics Library.
Victor Yohai, is a Professor in the Department of Mathematics, Faculty of Exact and Natural Sciences, University of Buenos Aires, Argentina, and researcher at CONICET. He is the author of a large number of important research articles on robust statistics, in particular on regression and time series. Several of the procedures proposed by him have been implemented in the robust library of SPLUS.
"…an original and valuable contribution…a source of inspiration for all those pursuing research in robust statistics." (Mathematical Reviews, 2007i)
"…a great book for graduate students as well as for applied scientists and data analysts." (MAA Reviews, February 14, 2007)
Robust Statistics: Theory and Methods (US $119.00)
and Linear Statistical Inference and its Applications, 2nd Edition (US $164.00)
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